Factual-Anchored Counterfactuals for Causal-Diagnostic Explanations

A counterfactual explanation of an observed case requires two choices: how alternative values are coupled to the values that actually occurred, and how much of the observed case the alternative world inherits downstream. We introduce factual-anchored counterfactuals (FACs), which encode these choices through natural-value-dependent laws. Scoped variables are revised by stochastic rules explicitly anchored at their realized factual values, while downstream propagation proceeds either through the original unit-level mechanisms (unit-persistent, u-FAC) or by sequential resampling from endogenous cross-world laws (resampling, r-FAC). We study diagnostic queries that condition simultaneously on the factual case and on a desired alternative outcome. Both variants admit the same Bayes decomposition into a factual-anchored revision law and a target-success score. However, they differ in their need for latent-background abduction. The u-FAC diagnostic law is generally not identified by the causal graph, revision law, and observed data. Conversely, r-FAC is point identified when its resampling laws are defined from the observed distribution, whilst giving up interventionist unit-level semantics. Finally, the revision law can be specified as an exponential tilt of the observed law, revealing connections between FAC-based explanations and feature-space, disturbance-based, and optimal-transport approaches to counterfactual explanation.